{"id":517,"date":"2012-12-10T22:06:26","date_gmt":"2012-12-10T13:06:26","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=517"},"modified":"2021-09-29T20:11:02","modified_gmt":"2021-09-29T11:11:02","slug":"thr200802","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr200802\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2008\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b.\r\n\u5e73\u9762\u4e0a\u306e \\(\\triangle \\text{OA} {} _ 1 \\text{A} {} _ 2\\) \u306f \\(\\angle \\text{OA} {} _ 2 \\text{A} {} _ 1 = 90^{\\circ}\\) , \\(\\text{OA} {} _ 1 = 1\\) , \\(\\text{A} {} _ 1 \\text{A} {} _ 2 = \\dfrac{1}{\\sqrt{n}}\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b.\r\n\\(\\text{A} {} _ 2\\) \u304b\u3089 \\(\\text{OA} {} _ 1\\) \u3078\u5782\u7dda\u3092\u304a\u308d\u3057, \u4ea4\u70b9\u3092 \\(\\text{A} {} _ 3\\) \u3068\u3059\u308b.\r\n\\(\\text{A} {} _ 3\\) \u304b\u3089 \\(\\text{OA} {} _ 2\\) \u3078\u5782\u7dda\u3092\u304a\u308d\u3057, \u4ea4\u70b9\u3092 \\(\\text{A} {} _ 4\\) \u3068\u3059\u308b.\r\n\u4ee5\u4e0b\u540c\u69d8\u306b, \\(k=4, 5, \\cdots\\) \u306b\u3064\u3044\u3066, \\(\\text{A} {} _ k\\) \u304b\u3089 \\(\\text{OA} {} _ {k-1}\\) \u3078\u5782\u7dda\u3092\u304a\u308d\u3057, \u4ea4\u70b9\u3092 \\(\\text{A} {} _ {k+1}\\) \u3068\u3057\u3066, \u9806\u756a\u306b \\(\\text{A} {} _ 5 , \\text{A} {} _ 6 , \\cdots\\) \u3092\u5b9a\u3081\u308b.\r\n\\(\\overrightarrow{h _ k} = \\overrightarrow{\\text{A} {} _ k \\text{A} {} _ {k+1}}\\) \u3068\u304a\u304f\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(k=1, 2, \\cdots\\) \u306e\u3068\u304d, \u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{h _ k}\\) \u3068 \\(\\overrightarrow{h _ {k+1}}\\) \u306e\u5185\u7a4d \\(\\overrightarrow{h _ k} \\cdot \\overrightarrow{h _ {k+1}}\\) \u3092 \\(n\\) \u3068 \\(k\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(S _ n = \\textstyle\\sum\\limits _ {k=1}^{n} \\overrightarrow{h _ k} \\cdot \\overrightarrow{h _ {k+1}}\\) \u3068\u304a\u304f\u3068\u304d, \u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n\\) \u3092\u6c42\u3081\u3088. \u3053\u3053\u3067, \u81ea\u7136\u5bfe\u6570\u306e\u5e95 \\(e\\) \u306b\u3064\u3044\u3066, \\(e = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1 +\\dfrac{1}{n} \\right)^n\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u3066\u3082\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n\"\"\r\n

    (1)<\/strong><\/p>\r\n

    \\(\\angle \\text{A} {} _ 1 \\text{OA} {} _ 2 = \\theta\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\sin \\theta = \\dfrac{1}{\\sqrt{n}}\r\n\\]\r\n\\(\\triangle \\text{A} {} _ {k+1} \\text{A} {} _ k \\text{A} {} _ {k+2} \\sim \\triangle \\text{OA} {} _ 1 \\text{A} {} _ 2\\) \u306a\u306e\u3067\r\n\\[\r\n\\left| \\overrightarrow{h _ {k+1}} \\right| = \\left| \\overrightarrow{h _ {k}} \\right| \\cos \\theta\r\n\\]\r\n\\(\\left| \\overrightarrow{h _ 1} \\right| = \\dfrac{1}{\\sqrt{n}}\\) \u306a\u306e\u3067\r\n\\[\r\n\\left| \\overrightarrow{h _ {k}} \\right| = \\dfrac{\\cos^{k-1}}{\\sqrt{n}}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{h _ k} \\cdot \\overrightarrow{h _ {k+1}} & = \\left| \\overrightarrow{h _ {k}} \\right| \\left| \\overrightarrow{h _ {k+1}} \\right| \\cos ( \\pi -\\theta ) \\\\\r\n& = -\\dfrac{\\cos^{k-1}}{\\sqrt{n}} \\cdot \\dfrac{\\cos^{k}}{\\sqrt{n}} \\cdot \\cos \\theta \\\\\r\n& = -\\dfrac{\\cos^{2k} \\theta}{n} \\\\\r\n& = -\\dfrac{\\left( 1 -\\sin^2 \\theta \\right)^k}{n} \\\\\r\n& = \\underline{- \\dfrac{1}{n} \\left( 1 -\\dfrac{1}{n} \\right)^k}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nS _ n & = -\\textstyle\\sum\\limits _ {k=1}^{n} \\dfrac{1}{n} \\left( 1 -\\dfrac{1}{n} \\right)^k \\\\\r\n& = -\\dfrac{1}{n} \\left( 1 -\\dfrac{1}{n} \\right) \\cdot \\dfrac{1 -\\left( 1 -\\frac{1}{n} \\right)^n}{1 -\\left( 1 -\\frac{1}{n} \\right)} \\\\\r\n& = -\\left( 1 -\\dfrac{1}{n} \\right) \\left\\{ 1 -\\left( 1 -\\dfrac{1}{n} \\right) \\left( \\dfrac{n-1}{n} \\right)^{n-1} \\right\\} \\\\\r\n& = -\\left( 1 -\\dfrac{1}{n} \\right) \\left\\{ 1 -\\left( 1 -\\dfrac{1}{n} \\right) \\cdot \\dfrac{1}{\\left( 1 -\\frac{1}{n-1} \\right)^{n-1}} \\right\\} \\\\\r\n& \\rightarrow -1 \\left( 1 -1 \\cdot \\dfrac{1}{e} \\right) \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} ) \\\\\r\n& = -1+\\dfrac{1}{e}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n = \\underline{-1+\\dfrac{1}{e}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \u5e73\u9762\u4e0a\u306e \\(\\triangle \\text{OA} {} _ 1 \\text{A} {} _ 2\\) \u306f \\(\\angle \\text{OA} {} _ 2 \\text … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[76],"tags":[148,16],"class_list":["post-517","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2008","tag-tohoku_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/517","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=517"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/517\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=517"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=517"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=517"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}